物理网络中局部学习规则的存储容量
Conservation capacity of local learning rules in physical networks
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中文总结 AI 辅助
本研究解决了物理网络局部学习规则的存储容量开放问题，基于特勒根定理建立容量理论，给出了扇区质量的拓扑计数方法，为电阻网络训练提供了理论基础。
中文摘要 AI 辅助
局部学习规则能保护多少个记忆？又是什么决定了这个数量？物理学习规则通过局部测量训练电阻网络，标准规则会守恒电导的类质量函数，该函数的一般形式曾作为开放问题提出，当时学界认为不存在有用的求解理论。我们给出了答案，且结果与预期相反，即存在一套容量理论。守恒质量背后的特勒根定理具有局域性：反馈状态在输入处被固定为零，因此输入电极分隔的电路的每个扇区都携带自身特有的守恒质量，该论证仅依赖基尔霍夫定律和该边界条件，对任意非线性分支定律均有效。独立扇区质量的数量是电路的拓扑性质，受输出电极数量的限制。不可训练元件会破坏其自身扇区的质量，而不会破坏其他扇区的质量，这指明了固定非线性元件的放置位置；每条边的学习率选择守恒的泛函，而非守恒的数量；伴随耦合学习（在测量前钳位其输出）会以输出平方乘数设定的速率耗尽每个带输出的扇区。在线性电路中，我们对相关定理进行分类：对多达5个顶点、2个输入和1或2个输出的全部502个电路以及数百个更大的电路进行精确有理计算，得到了闭式计数，该计数在该电路族上已被证明，且在一般情况下仍为猜想，扇区相关陈述在所有电路规模下均为定理。预期的串并联多项式定律恰好表现为串类三次差值，而开篇问题所问的数量是一种预算：每个扇区对应一个设计的质量，差值由拓扑提供，每个受保护的泛函对应一个广播标量。
英文摘要
How many memories can a local learning rule protect, and what fixes the number? Physical learning rules train resistive networks through local measurements, and the standard rules conserve a mass-like function of the conductances, a law whose general form was posed as an open problem with the expectation that no useful solution theory exists. We answer it, in the direction the expectation ran against, and the answer is a capacity theory. The Tellegen identity behind the conserved mass localizes: the feedback state is pinned to zero at the inputs, so every sector of the circuit that the input electrodes separate carries its own private conserved mass, by an argument consuming only Kirchhoff's law and that boundary condition, hence valid for arbitrary nonlinear branch laws. The number of independent sector masses is a topological property of the circuit, bounded by the number of output electrodes. An untrainable element can destroy the mass of its own sector and of no other, which says where fixed nonlinear components may be placed; per-edge learning rates select which functionals are conserved and never how many; and adjoint coupled learning, which clamps its outputs before measuring, drains every output-carrying sector at a rate set by the squared output multipliers. In linear circuits we then classify the identities: exact rational computation over all 502 circuits on up to five vertices with two inputs and one or two outputs, and hundreds of larger ones, yields a closed-form count, proved on that family and conjectured in general, with the sector statement a theorem at every circuit size. The anticipated series-parallel polynomial laws appear as exactly the series-class cubic differences, and the number the opening question asks for is a budget: one designed mass per sector, the differences the topology donates, and one broadcast scalar for each protected functional beyond.
